HAL Id: jpa-00247777
https://hal.archives-ouvertes.fr/jpa-00247777
Submitted on 1 Jan 1992
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Flow enhancement of electrical fluctuations in polymer solutions and melts
Helmut Brand, Harald Pleiner
To cite this version:
Helmut Brand, Harald Pleiner. Flow enhancement of electrical fluctuations in polymer solutions and melts. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.1909-1914. �10.1051/jp2:1992109�.
�jpa-00247777�
Classification
Physics Abstracts
61A0K 05.70L 77.90
Short Communication
Flow enhancement of electrical fluctuations in polymer solutions
and melts
Helmut R. Brand and Ilarald Pleiner
FB 7, Physik, Universitit Essen, D-4300 Essen I, Germany
(Received 24 June1992, accepted in filial iofin 31 August 1992)
Abstract. We show that the origin of the enhanced electrical fluctuations under flow, ob- served over the last is years in polymer solutions, can be traced back to a dissipative cross coupling between the strain tensor (characterizing transient polymeric networks) and electric
fields. Additional experiments to further test our picture are suggested.
1. Iuti~oduction.
In a series of papers over the last fifteen years Ilublt and his group [1-7] have reported a
number of features of a fascinating phenomenon, which has not been explained or cast into
a proper framework up to now. In polymer solutions they have observed an enhancement of electrical fluctuations during flow when compared to those at rest. Without an external flow field the intensity of the electrical fluctuations corresponds to the value expected from
thermodynamics [8, 9], and is in this case given by the Nyquist formula [10]. This situation
changes drastically as soon as a flow of sufficient strength is imposed on the polymer solutions in question ([or example, polyethylene oxide solutions). While we will not attempt here to
explain the phenomena observed in all detail, especially not the nonlinear aspects, we think that the approach presented by us [IIi in this journal recently, namely the linearized macroscopic dynamics of polymer solutions and melts, can provide a sensible framework to describe the
experimental results. Our description is unique when compared to other approaches to polymer solutions, since it allows in a natural way to incorporate couplings between the transient
network characterized by the strain tensor e;j and other macroscopic variables such as the temperature variations ST, the concentration c and the electric field E.
1910 JOURNAL DE PHYSIQUE II N°11
2. Expei~imental situation.
We first summarize briefly some important aspects of the experimental results []-ii. As it has been found consistently in all experiments reported, there is a minimum flow rate, I-e-
a threshold above which the electrical fluctuations are enhanced. In addition it has been observed that a certain degree of "elasticity" of the polymer solutions is necessary to obtain
an
enhancement. Very crudely speaking one could characterize this degree of "elasticity" by the relaxation constant r2 of the transient network. For very small r~, as for example in very dilute solutions, the enhancement of the electrical fluctuations observed is negligible, while for larger
values of r2 a strong enhancement is observed. In this picture the limit r2
- cc corresponds to a
genuine solid. In the experiments it has also been found that small scale flow structures, which
occur as vortices or even turbulent flows near the inlet region of the capillaries, are important
in getting a large elTect. Thus, small length scales or large wavevectors are favourable for a
big enhancement of the electrical fluctuations. In addition to these qualitative features just outlined, the experiments have revealed a number of phenomena that are clearly linked to strictly nonlinear properties.
Here we want to demonstrate that all the qualitative features outlined above result naturally
from our macroscopic dynamic approach [11], Which we have generalized meanwhile to liquid crystalline polymers [12-14] possessing additional degrees offreedom such as orientational order.
3. Eleiuents of the macroscopic dynai»ic appi~oach and its application to the
experiments,
In macroscopic dynamics one keeps firstly all truly hydrodynamic degrees of freedom, which do not relax in the infinite wavelength limit [15, 16], and secondly those macroscopic degrees
of freedom, which relax on a finite but sufficiently long time scale, I-e- long compared to the
microscopic time scales. The classical example for this type of variable is the modulus of the order parameter near second order phase transitions such as for example near the I transition in ~He (compare Ref. [17]). Another example for
a macroscopic variable in certain condensed systems (e.g. in complex fluids including liquid crystals) is the macroscopic electric polarization
P [18]. In reference [I ii we have generalized further this approach and incorporated the strain tensor e;j as a macroscopic variable to account for the fact that in polymer solutions and melts there exists
a "transient network" [19, 20], I-e- for sufficiently small frequencies the polymer
behaves like a simple liquid whereas for sufficiently large frequencies it reacts like a solid.
Frequently this phenomenon is called the dynamic glass transition [21].
In this approach the additional dynamic equation for the strain tensor eij is coupled to the other macroscopic variables, namely to the conserved quantities density p, energy density e, density of linear momentum g and concentration c as well as to the macroscopic polarization
P making use of general symmetry arguments [15, 16] and irreversible thermodynamics [8].
The former include invariance under translations and rotations, Galilean invariance and the appropriate symmetries under parity and time reversal. In reference [iii we have given and discussed in detail the linearized macroscopic dynamic equations and we therefore concentrate in the following only on those aspects which are helpful in providing a framework for the
experimental observations by I(ublt's group. As we have pointed out there are static and dynamic couplings between the strain tensor e;j and gradients of the electric field, where the latter coupling will turn out to be of crucial importance in our interpretation of the experimental results.
For the dynamic equations of the density of linear momentum, the strain tensor and the
polarization we have, including fluctuations
j; + Via;j + Vi L;j = 0 (1)
I;j + X;j + f;j = 0 (2)
Pi + if + Jf
" ° (3)
where the fluctuating currents L;j, f;j, and J~~ are introduced into the macroscopic dynamic equations along the lines pioneered by Landau and Lifshitz [22]. The reversible parts of the
deterministic currents a;j, Xij, and jf are [iii
a(
= p6;j 4lij (4)
X(
= -A;j (5)
Jf~
" o (6)
where 4l;j is the thermodynamic conjugate to the strain tensor e;j and where Ajj = )(T7;vj +
T7jv;) with the velocity field vi. The relevant part of the dissipation function R reads
2R =
j dr[.
+ );jki4l;j 4lki + a~E~ + 2(~E;
T7j4lij] (7)
r
where indicate terms not relevant below, which are listed, together with the explicit form of the material tensors, in reference [11]. From the dissipation function we get the dissipative
currents in equations (4-6) by taking a variational derivative of R with respect to the forces, for example
jjD
~
6R ~~~ ~
~~' ~ ~~ ~° (8)
The fluctuating forces are linked to the dissipative transport parameters via the fluctuation dissipation theorem [8, 9, 23]
< l~iJ(r>t) l~ki(r'>t') >
" 2kB T ~iJki b(r r') b(t t') (9)
< IQ(i~,t)fLi(i~',t') >
= 2kB Tii);)
ki 6(r r') 6(t t') (10)
< Jj(i~, t)Jj(i~',t') >
= 2kB T al
6;j 6(r r') 6(t t') (II)
< IQ (r,t) Jj(i~',t') >
= 2kB T(~ 61k T7j 6(r r') 6(t t'). (12)
Combining equations (1-8) we find for the linearized macroscopic dynamic equations of the strain tensor eij and the macroscopic polarization l§
iij Aij + ())ijki 4lki (~Vi E; = 0 (13)
Pi + a~E, + (~ Vi4l;j
= 0. (14)
From equations (13),(14) we read off immediately that the symmetrized velocity gradient A;j
drives the strain tensor eij and this in turn gives rise to variations in the polarization Pi via the dissipative cross coupling (~ Since the strength of the velocity gradients and their fluctuations have not been measured quantitatively in the experiments described so far (cf. the discussion below), we will just take the intensity of the velocity variations as an input parameter and
denote it by A.
1912 JOURNAL DE PHYSIQUE II N°li
It is then straightforward using standard techniques [16] to get from equations (13), (14) the spectrum of the electric field fluctuations as a function of the wavevector k and the frequency
w. For the spectrum in the presence of flow we obtain after Fourier transformation in space and time, assuming incompressibility and neglecting crosscoupling effects in the denominator
~ (~~2(E)2 ~
~ ~~~ ~~~~
~J~(X/C2)~ + W~[(4X/T2)~ + l'~/C2)~] + (4'~/T2)~ ~~~~
where x is the electric susceptibility and c~ the elastic shear modulus [24]. This expression
must be compared with the spectrum of thermal fluctuations of the electric field in the absence of flow, which is given neglecting crosscoupling effects by
< jEj2 >~~=
'~
(Wx)~ + l'~)~' (~6)
Taking the ratio of these two expressions we get the enhancement of the electric fluctuations without further siInplifications
~ ~~2~~
~~~~~~~~~~
w2
~~~/r2)~
~~~~
Several remarks are to be made:
I) a large wavevector is favourable for getting a big enhancement, which is proportional
to k~;
it) the enhancement is proportional to the square of the dissipative cross coupling (~ intro- duced in reference [11];
iii) taking the low frequency liiuit we also see that the enhancement is proportional to the square of r2 characterizing the degree of "elasticity" of the polymer solution or melt;
iv) from a more general analysis, including all crosscoupling effects in equations (15) and
(16), it follows that in equation (17) a~ is replaced by a~ + x F(z) with x % ((~k)2, where
F(x) depends in a very complicated manner on various hydrodynamic parameters as well as
on the frequency w and the wavevector k;
v) within our linearized description the enhancement is proportional to the intensity of the
velocity variations. However, nonlinear effects are likely to change this proportionality and, in addition, the dependence on the wavevector and on the cross coupling (~ to noninteger powers.
Nevertheless the following picture emerges: an inhomogeneous hydrodynamic flow generates
a strain tensor e;j in polymer solutions with a sufficiently large degree of "elasticity". This strain increases if the lengthscale of the flow decreases. Such small scale, large wavevector flows can be obtained experimentally, for example, above the onset of an instability. This step has been suggested in a slightly difserent wording already in references []-ii. But the decisive step to have a framework into which the observed effects can be incorporated is to realize that a transient network in the presence of small scale spatial variations gives rise invariably
to enhanced electrical fluctuations via the dissipative cross coupling (~, whose existence has been pointed out first in reference [11]. From our analysis we see clearly the advantage of
large wavevectors. iiTe would like to stress that one would not know how to incorporate such
a cross coupling into iiiore conventional descriptions of polymer flows [19]. From the approach
of macroscopic dynamics, hoc,ever, such cross-couplings emerge quite naturally.
4. Suggested future expei~iiuents.
From the analysis given above quite a few possibilities for future experiments emerge immedi- ately. First of all it would be very important to vary the diameter of the capillaries used to
determine the precise dependence of the noise enhancement on the characteristic length scale of the flow. A second issue which clearly deserves further analysis is the measurement of the fluctuation spectrum of the velocity field. If the picture suggested is applicable, an enhance- ment of the intensity of the velocity variations should also give rise to an enhancement of the
intensity of the electric fluctuations. From equations (13) and (14) it emerges that a time trace of the velocity and of the electric field are related. While these equations are strictly applicable only in the linearized domain, one might expect that even in the nonlinear regime the time
series of velocity and electric field should show many qualitative similarities.
Finally we would like to point out that dissipative cross coupling terms to the strain tensor, similar to that of the electric field, also exist for temperature and concentration gradients. Thus, together with the fluctuations in the electric field, those in temperature and concentration are also generated by the small scale flo~v. ~vhile it might be difficult to detect the enhancement of the temperature fluctuations due to the cross coupling, because they are probably masked by
Ohmic and viscous heating due to the flow, it could be possible to detect an enhancement in the concentration fluctuations of the polymer solutions studied, provided one finds
a convenient experimental way to measure variations of the chemical potential.
5. Conclusions and pei~spectives,
In this note we have suggested a fi.amework to account for experimental observations, which give
enhanced electrical fluctuations in flowing polymer solutions when compared to thermodynamic
fluctuations. ~Ve have traced this enhancement back to a dissipative cross coupling between the strain tensor associated with the transient polymeric network (generated by the small scale flow) and spatial variations of the electric field. This picture arises naturally only in the approach of macroscopic dynamics for polymer solutions, but is absent in the more conventional
descriptions of polymer dynamics. ~~e have also suggested several experiments to further test the given picture. In the future we hope to analyze the intrinsically nonlinear phenomena
observed in the experiiuents.
Acknowledgemeiits.
It is a pleasure to thank J. Ilub6t and O. Quadrat for useful correspondence.
Support of this work by the Deutsche Forscliungsgemeinschaft is gratefully acknowledged.
References
[1] KLASON C., KUBiT J., J. Appl. Phys. 47 (1976) 1970.
[2] HEDMAN Il., I(LASON C., IIUBiT J., J. Appl. Phys. 50 (1979) 8102.
[3] HEDMAN Il., I(LASON C., POUPETOVA P., Rheol. Acta 22 (1983) 449.
[4] HEDMAN Il., IIUBiT J., Rheol. Acta 23 (1984) 417.
[5] KLASON C., I(UBiT J., J. Appl. Phys. 58 (1985) 2040.
1914 JOURNAL DE PHYSIQUE II N°11
[6] HEDMAN K., IIUBiT J., Rheol. Acta 25 (1986) 36.
[7] KLASON C., IIUBiT J., QUADRAT O., Polymer 33 (1992) 1464.
[8] DE GROOT S-R-, MAZUR P., Nonequilibrium Thermodynamics (North Holland, Amsterdam, 1962).
[9] LANDAU L-D-, LIFSHITZ E-M-, Statistical Physics (Pergamon Press, New York, 1959).
[10] NYQUIST H., Phys. Rev. 32 (1928) l10.
[II] BRAND H-R-, PLEINER H., RENZ W., J. Phys. France 51 (1990) 1065.
[12] PLEINER H., BRAND H-R-, Mol. Cryst. Liq. Cryst. 199 (1991) 407.
[13] PLEINER H., BRAND H-R-, Jfacromolecltles 25 (1992) 895.
[14] PLEINER H., BRAND H-R-, Polym. Adu. Techn. 3 (1992).
[15] MARTIN P-C-, PARODI O., PERSHAN P-S-, Phys. Rev. A6 (1972) 2401.
[16] FORSTER D., Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Ben- jamin, Reading, Mass., 1975).
[17] KHALATNII(OV I-M., Introduction to the Theory of Superfluidity (Benjamin, New York, 1965).
[18] de GENNES P-G-, The Physics of Liquid Crystals (Clarendon Press, Oxford, 3rd edition, 1982).
[19] FERRY J-D-, Viscoelastic Properties of Polymers (Wiley, New York, 2nd edition, 1970).
[20] de GENNES P-G-, PINCUS P-A-, J. Chimie Phys. 74 (1977) GIG.
[21] GRAESSLEY W.W., Adv. Polym. Sci. 16 (1974) 1.
[22] LANDAU L-D-, LIFSHITZ E-M-, Hydrodynamics (Pergamon Press, New York, 1959).
[23] Equation (12) corrects a typographical error in equation (A2) of reference [I1].
[24] Since we assumed incompressibility, the bulk elastic modulus cl (and ri did not occur in equation (is). We now believe that an additional restriction on cl and c2 (Eq. (2.13) in [iii) is
unnecessary (cf. Ref. [12] for a detailed discussion).
